Speculative Attacks and the Theory of Global Games
Frank Heinemann, Technische Universität Berlin
Barcelona LeeX Experimental Economics Summer School in Macroeconomics
Universitat Pompeu Fabra
1
Coordination games with multiple Equilibria
Currency Attack (Obstfeld, 1986, 1996, 1997)
If traders expect devaluation, they sell currency.
Increased supply generates pressure to devaluate.
In critical cases, pressure gets large enough for central bank to devaluate,
even if it had kept the current rate without this pressure.
Devaluation
unavoidable
Multiple equilibria
No devaluation
Y fundamentals
Y
Y
2
Firm Liquidation
Borrowers expect liquidation and therefore withdraw credit
 Higher
 in
costs of capital
critical cases: Bankruptcy that could have been avoided without higher costs of capital
even if it had kept the current rate without this pressure.
liquidation
unavoidable
Multiple equilibria
No liquidation
Y fundamentals
Y
Y
Other coordination games with similar structure:
Bank runs
Competition between intermediaries
Competition between networks
3
Multiple equilibria
= Self fulfilling beliefs
= Existence of sunspot equilibria
Financial system is prone to sudden shifts in beliefs that may trigger a crisis.
Shifts in beliefs do not need to be related to news on fundamentals.
Inherent instability
Financial Crises are (to some extend) unpredictable
Impossibility to analyse comparative statics
No clear policy recommendations
Other viewpoint:
The model has multiple equilibria, because it lacks an important determinant of decisions
4
A Speculative-Attack Model
Literature: Morris / Shin (1998), Heinemann (2000)
Nature selects
Fixed exchange rate
Shadow exchange rate
Y
e*
f(Y)
random, uniform in [0,1]
measured in foreign curr. / domestic curr.
f’ > 0
Higher state Y = better state of the economy
2-stage game:
1. Private agents decide on whether or not to attack the currency. Continuum of agents i  [0,1]
Attack = sell domestic currency. Each agent can sell one unit.
Let α be the proportion of agents who sell the currency (measure of speculative pressure).
2. Central bank gives up the currency peg if and only if proportion of attacking agents is larger
than a hurdle
a(Y ) .
If attack is successful, attacking agents earn R(Y) – t = e* – f(Y) – t.
If attack fails, attacking agents lose t.
5
α
„Hurdle function“
a(Y)
Give up the peg
a’ ≥ 0
Keep the peg
Y
Y
Assume: 1. there is a state Y  0 with a (Y )  0
2. there is a state Y  1 with R (Y )  t
=> for Y  Y the CB gives up anyway.
(or a (Y )  1 )
3. Y  Y
6
1
a(Y)
Y
0
Y
For Y  Y , attacking is
never rewarding
=> dominant strategy
not to attack
$
R(Y)
t
Y
0
1
Y
attack
Multiple equilibria
For Y  Y , attacking is
always successful and
rewarding.
=> dominant strategy
to attack
For Y  Y  Y , an
attack is rewarding
if and only if at least
a proportion of a(Y)
traders attack.
No attack
Note that the state of the world and the shadow exchange rate are common knowledge.
In any Nash equilibrium, strategies are also common knowledge!
7
Global Game Approach
(Carlsson / van Damme (1993), Morris / Shin (1998, 2003), Heinemann (2000))
Nature selects
Y
random
Players get private information
Xi = Y + ui
random terms ui are i.i.d.
Assumption: Conditional variance of private signals Var (Xi | Y) is sufficiently small

Unique equilibrium with threshold X*, such that
players with signals Xi < X* do not attack,
players with signals Xi > X* do attack.
Marginal player with signal X* is indifferent.
Additional Equilibrium condition:
E Ui (attacking | X*) = E Ui (non attacking | X*)
8
Global Game
Treat the model as a randomly selected model out of a class of models
True state of the world Y is random, Y ~ uniform in [0,1]
Players do not know Y.
Players get private information
ui ~ uniform in [ – ε, + ε],
xi = Y + ui
random terms ui are i.i.d.
ε is small,   min{Y / 2, (1  Y ) / 2}
The game is supermodular
Supermodular games have a highest and a lowest equilibrium that can be attained by iterative
elimination of dominated strategies (Vives 1990, Milgrom/Roberts 1990).
Apply iterative elimination of dominated strategies => iteration procedures from above and below
stop at threshold strategies, such that any player i attacks, if her signal is smaller than the threshold,
does not attack if her signal is larger, and is indifferent if her signal equals the threshold.
9
Let us look at some threshold strategy x*. A player i attacks if and only if
The proportion of agents attacking is
0

 x *   Y
s(Y , x*)  
2

1

xi  x * .
if
Y  x * 
if
x *   Y  x * 
if
Y  x * 
1
s(Y,x*)
0
x*–ε
x*
x*+ε
Y
10
Let us look at some threshold strategy x*. A player i attacks if and only if
The proportion of agents attacking is
0

 x *   Y
s(Y , x*)  
2

1

1
xi  x * .
if
Y  x * 
if
x *   Y  x * 
if
Y  x * 
a(Y)
s(Y,x*)
0
x*–ε
x*+ε
Y
Y*(x*)
Attack successful
Attack not successful
11
Is this an equilibrium?
i
i
Player i prefers attacking if and only if x  x * . If x  x * the player is indifferent
Additional equilibrium condition: E Ui (attacking | x*) = E Ui (non attacking | x*)
See Figure 1
Uniqueness?
See Figure 2
Generally, uniqueness depends on distribution.
For uniform distribution, we get a unique equilibrium if ε is small.
For normal distribution, uniqueness requires that Variance of private signals xi|Y is small compared
to variance of prior distribution of Y.
Limit case: ε → 0 => x* = Y*, determined by
R(Y *) (1  a(Y *))  t .
12
Global Game Approach – Some Theoretical Results:
1.
Uniform distribution of signals => Equilibrium is unique (Carlsson/van Damme 1993,
Morris/Shin 1998, 2003)
2.
For ε→0, critical state Y* converges to Y0, solution of (1-a(Y0)) R(Y0)= t
(Heinemann 2000). Call this limit point the Global Game Selection (GGS)
3.
Y ~ N (y,τ2) ,
Xi = Y + ui ,
ui ~ N (0,σ2)
2
Equilibrium is unique, provided that    a' (Y ) 2
for all Y.
(Morris/Shin 1999, Hellwig 2002)
4.
In a game with just two actions (attack or not): GGS is independent of particular
distribution (noise-independent) and can be characterized by best response of a
player who believes that the proportion of others attacking has a uniform distribution
in [0,1]. (Frankel et al. 2001)
5.
Uniform distribution: transparency (lowering ε) reduces prior probability of crises
(Heinemann/Illing 2002) see Figure 4.
6.
Normal distribution: Ambiguous effects of public and private info (Rochet/Vives 2002,
Metz 2002, Bannier/Heinemann 2005).
A well informed large trader increases probability of crises (Corsetti et al. 2004)
13
7.
If a game with more than 2 actions can be decomposed in smaller noise independent
games, and the GGS of these smaller games all point towards the same strategy,
then this strategy is also the noise independent GGS of the larger game
(Basteck, Daniëls, Heinemann, 2013).
Example: n players, m actions: a1, a2, a3, …, am, with : aj < aj+1 for all j
Look at the games with the same set of players but only 2 adjacent actions aj, aj+1.
If there is some j*, s.t. for all restricted games (aj, aj+1) with j<j*, the GGS is aj+1, and
for all games (aj, aj+1) with j≥j*, the GGS is aj, then aj* is the noise independent GGS
of the large game.
Since GGS of 2-action games can be easily calculated, you can also easily calculate
the GGS of a noise independent larger game.
If a game is not noise independent, multiple equilibria of CK game are replaced by
multiple GGS (depending on distribution). Here the concept is less convincing.
14
Interpretation of GGS as a refinement theory for common knowledge game:
If variance of private signals approaches zero, in equilibrium each player acts as if she
beliefs that the proportion of other players attacking has a uniform distribution in [0,1].
(Laplacian Beliefs)
 Global Game Selection
Other refinement theories
- payoff dominant equilibrium
- risk dominant equilibrium
- maximin strategies
15
Experiment on the speculative attack game
(Heinemann, Nagel, Ockenfels 2004)
1. People use threshold strategies (as predicted by global games)
2. Public information is not destabilizing
3. Comparative statics w.r.t. parameters of payoff function as predicted
by global-games theory
4. Subjects coordinate on strategies that yield a higher payoff than
global-game equilibrium (thresholds are lower than predicted)
16
Sessions with common information and sessions with private information
15 participants per session (undergraduate students of Goethe Universität Frankfurt and
Universitat Pompeu Babra in Barcelona)
Each session consists of 16 rounds,
each round has 10 independent decision situations:
Decision situation:
All sessions:
Y  [ 10, 90 ]
randomly selected (uniform distribution)
Sessions with private information only:
Xi | Y  [ Y – 10 , Y + 10 ]
independently and randomly selected (uniform distribution)
Subjects decide between A and B
- knowing Y in sessions with common information
- knowing Xi in sessions with private information
17
Payoff for A is certain and either T=20 or T=50 ECU (Experimental Currency Units)
We have two treatments per session, each is kept for 8 rounds:
Half of the sessions start with T=20 for the first 8 rounds and switch to T=50 thereafter.
In other sessions: reverse order
Payoff for B is Y, if at least
a(Y) = 15 * (80-Y) / Z
subjects choose B, zero otherwise.
”risky action”
(In four sessions Z=100, in others Z=60)
B may be interpreted as “Attack”.
Y
difference between currency peg and shadow
exchange rate (larger Y = worse economic state)
a(Y) amount of capital needed to enforce devaluation
A may be interpreted as “non-Attack”.
T opportunity costs of an attack
18
Hurdle to Success
15
12
a(Y) for Z=60
a(Y) for Z=100
9
6
3
0
Y
0
10
20
30
40
50
60
70
80
90
100
19
To calculate the minimal number of B-players needed to get a positive reward for B,
subjects get the following table (sessions with Z=60):
If the unknown number Y is in the interval,
(Note: Y is between 10 and 90)
20,00 to 23,99
24,00 to 27,99
28,00 to 31,99
32,00 to 35,99
36,00 to 39,99
40,00 to 43,99
44,00 to 47,99
48,00 to 51,99
52,00 to 55,99
56,00 to 59,99
60,00 to 63,99
64,00 to 67,99
68,00 to 71,99
72,00 to 75,99
76,00 to 90,00
Then at least … of the 15 participants (including yourself)
must select B, in order to get a positive payoff
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
20
Information phase:
Each subject gets to know for each of the 10 decision situations
- the number Y
- how many subjects chose B
- how her payoff changed by her decision.
Payments:
In sessions with Z=100: 1000 ECU = 4 DM
In sessions with Z = 60: 1000 ECU = 5 DM (Frankfurt) or 300-400 Ptas. (Barcelona)
Average payments between 14.30 and 22.50 €, duration 90-120 minutes per session.
Additional sessions:
40 periods with T = 50: 1000 ECU = 1 €
High stake sessions
1 ECU = 1 € for one selected situation from each of the 2 stages
21
Participants:
Students from Goethe-University Frankfurt am Main and from University Pompeu Fabra in
Barcelona, mainly economics undergraduates, invited by leaflets, posters and an e-mail to all
students with account at the economics department in Frankfurt.
Place:
Experiment were run in a PC-pool with separated working spaces in Frankfurt and in the LEEX
laboratory in Barcelona.
Software:
We used z-Tree, developed by Urs Fischbacher (University of Zürich).
22
Session Overview
Number of sessions with
Z
Secure payoff T
Location
100
1st stage 20
2nd stage 50
Frankfurt
100
50 / 20
60
session type
Public information
Private information
standard
1
1
Frankfurt
standard
1
1
20 / 50
Frankfurt
standard
1
2
60
20 / 50
Barcelona
standard
3
3
60
50 / 20
Frankfurt
standard
2
2
60
50 / 20
Barcelona
standard
3
3
Control sessions:
60
20 / 50
Frankfurt
experienced
1
60
50 / 20
Frankfurt
experienced
1
60
50
Barcelona
40 periods
2
60
20 / 50
Barcelona
high stake
1
60
50 / 20
Barcelona
high stake
Total number of sessions
1
14
15
23
number of traders
a(Y) hurdle to successful attack
1
payoff, costs
Y payoff to successful attack
transaction costs
T
T
no attack
Y
multiple equilibria
state Y
attack
Figure 1. The speculative attack game. If at least a(Y) traders attack, attacking traders receive a payoff YT. Otherwise they loose T.
24
Game with common information (CI)
Payoff dominant equilibrium:
Choose A for Y < T and B for Y > T
Maximin strategy:
Choose A for Y < Y and B for Y >
Y
Global Game Solution:
Choose A for Y < Y0 and B for Y > Y0
T < Y0 <
Y
Game with private information (PI)
Unique equilibrium:
Choose A for Xi < X* and B for Xi > X*
T < X* <
Y
25
Theoretical Predictions
Treatments T=20, Z=100
T=20, Z=60
T=50, Z=100
T=50, Z=60
Equilibrium Theories
CI game
Payoff dominant equilibrium
T
20
20
50
50
Maximin equilibrium
Y
73.33
76.00
73.33
76.00
Global Game solution
Y0
33.33
44.00
60.00
64.00
Risk dominant equilibrium
34.55
44.00
62.45
67.40
Limiting logit equilibrium
33.07
48.00
51.48
56.00
32.36
41.84
60.98
66.03
PI game
Unique equilibrium
X*
Table 2. Theoretical equilibrium threshold states or signals for the parameters T and Z, and n=15.
26
Results:
1. Threshold Strategies
More than 90% of all subjects played threshold strategies.
In last round at least 13 out of 15 subjects played threshold strategies.
Theory:
In PI-game equilibrium strategies are threshold strategies.
In CI-game there are many other equilibria.
Evidence:
There is no significant difference between proportion of threshold strategies in sessions
with PI and CI.
Explanation: Non-threshold strategies are not robust against strategic uncertainty
27
Evolution of Threshold Strategies
treshold strategies
100%
90%
80%
private information
70%
common information
60%
1
2
3
4
5
6
7
8 9 10 11 12 13 14 15 16
period
Figure 4. Percentage of inexperienced subjects, whose behavior was consistent with undominated threshold
strategies.
28
2. Evolution of Coordination
Two kinds of mistakes: 1. choose B and receive zero (failed attack)
2. choose A when B would have been successful (missed opportunity to attack)
Evolution of Coordination
percentage of mistakes
20%
16%
12%
8%
4%
0%
1
2
3
4
5
6
7
8
9
mistakes in sessions with PI
10
11
12
13
14
15
16
Period
mistakes in sessions with CI
The difference is about the size that can be explained by incomplete information in PI-setting.
29
3. Thresholds to Crises
#B
Session 010201 - Stage 1: Z=60 PI T=20
15
12
9
6
3
0
10
20
30
40
50
60
70
80
90 Y
number of players choosing B
hurdle function for success a(Y)
largest Y without success for B
smallest Y with success for B
47,72
45,36
Figure 5. Data from one session and stage show 80 randomly selected states and associated numbers of subjects who
chose B. All data points on or above the hurdle function are successful attacks.
30
#B
Session 0607P9 - Stage 1: Z=60 CI T=50
15
12
9
6
3
0
10
20
30
40
50
60
70
80
90
Y
number of players choosing B
hurdle function for success a(Y)
largest Y without success for B
smallest Y with success for B
54,67
56,26
Figure 2. Combined data from all eight periods of one stage of a session with common information. In this example there is
complete separation of states with failed and successful attacks.
31
4. Estimation of average thresholds
Estimation of a logistic distribution that approximates data of the last four rounds in each treatment
Prob (B) = 1 / ( 1+exp(a-bx) )
Estimate parameters a and b, using logistic regression
estimated mean threshold
a/b
estimated standard deviation π / (b
3
)
32
Session 010201 E1
prob(B)
estimated mean threshold a/b
1
Surrounding of 1 estimated
standard deviation
decisions
log. estimation
0,5
X
Y
0
0
10
20
30
40
50
46,40
60
70
80
90
100
Session with private Information
33
proportion of
B
1
Session C3 CI Z=60 order=50/20 T=50
decisions
log. estimation
hurdle
0,5
0
10
20
30
40
50
60
70
80
90
Y
estimated mean threshold a/b = 57.09
Session with common information
34
Session
Type
Location
Z
Information
Order
T
P1
standard
Frankfurt
100
PI
20/50
P2
standard
Frankfurt
100
PI
50/20
C1
standard
Frankfurt
100
CI
20/50
C2
standard
Frankfurt
100
CI
50/20
P3
standard
Frankfurt
60
PI
20/50
P4
standard
Frankfurt
60
PI
50/20
P5
standard
Frankfurt
60
PI
50/20
P6
standard
Frankfurt
60
PI
20/50
C3
standard
Frankfurt
60
CI
20/50
C4
standard
Frankfurt
60
CI
50/20
C5
standard
Frankfurt
60
CI
50/20
P7
standard
Barcelona
60
PI
20/50
P8
standard
Barcelona
60
PI
50/20
P9
standard
Barcelona
60
PI
20/50
20
50
50
20
20
50
50
20
20
50
50
20
50
20
20
50
20
50
50
20
50
20
20
50
50
20
20
Parameter
estimation
a
b
5.07
0.155
11.13
0.196
12.78
0.237
9.88
0.370
10,32
0.311
67.43
1.265
10,37
0.198
15.16
0.750
5.67
0.123
7.15
0.119
7.85
0.134
7.29
0.157
12.79
0.211
11.92
0.289
7.40
0.166
18.37
0.305
9.13
0.239
36.28
0.635
8.08
0.177
10.32
0.314
330.25
6.402
14.24
0.443
7.94
0.185
7.82
0.144
14.09
0.264
10.52
0.291
7.51
0.167
Estimated
mean
a/b
32.76
56.77
53.90
26.71
33.21
53.31
52.37
20.22
46.04
60.32
58.59
46.57
60.71
41.22
44.57
60.29
38.20
57.09
45.67
32.81
51.58
32.16
42.84
54.16
53.35
36.18
44.86
Estimated standard
deviation
11.72
9.25
7.65
4.90
5.84
1.43
9.16
2.42
14.73
15.30
13.53
11.59
8.61
6.27
10.93
5.95
7.59
2.85
10.25
5.77
0.28
4.10
9.79
12.57
6.87
6.24
10.83
35
P 10
standard
Barcelona
60
PI
50/20
P 11
standard
Barcelona
60
PI
20/50
P 12
standard
Barcelona
60
PI
50/20
C6
standard
Barcelona
60
CI
20/50
C7
standard
Barcelona
60
CI
50/20
C8
standard
Barcelona
60
CI
20/50
C9
standard
Barcelona
60
CI
50/20
C 10
standard
Barcelona
60
CI
20/50
C 11
standard
Barcelona
60
CI
50/20
E1
Frankfurt
60
CI
20/50
Frankfurt
60
CI
50/20
L1
L2
H1
experienced
experienced
long
long
high stake
Barcelona
Barcelona
Barcelona
60
60
60
PI
PI
PI
–
–
20/50
H2
high stake Barcelona
60
CI
50/20
E2
50
50
20
20
50
50
20
20
50
50
20
20
50
50
20
20
50
50
20
20
50
50
20
50
50
20
50
50
20
16.68
10.32
9.88
8.08
14.82
13.45
8.02
6.33
11.35
23.33
17.61
25.71
73.82
8.75
14.36
6.31
10.11
21.36
17.59
18.19
28.83
85.88
16.09
22.78
19.96
8.38
10.12
37.79
46.56
0.326
0.188
0.259
0.188
0.247
0.237
0.231
0.162
0.223
0.430
0.490
0.639
1.356
0.158
0.340
0.154
0.176
0.411
0.477
0.557
0.505
1.707
0.518
0.378
0.357
0.148
0.156
0.668
6.051
51.24
55.00
38.14
43.09
60.01
56.73
34.78
39.10
50.87
54.25
35.96
40.26
54.44
55.49
42.22
40.94
57.50
51.91
36.92
32.66
57.06
50.32
31.06
60.36
55.96
56.79
65.07
56.58
46.56
5.57
9.66
7.00
9.67
7.35
7.65
7.86
11.20
8.13
4.22
3.70
2.84
1.34
11.50
5.33
11.77
10.31
4.41
3.81
3.26
3.59
1.06
3.50
4.81
5.09
12.29
11.66
2.72
6.05
36
5. Analyzing observed average thresholds
5.1. summary statistic
A Crisis occurs whenever Y > Y*. Lower threshold to crises <=> higher probability of crises
Treatment
T=20, Z=100
T=20, Z=60
T=50, Z=100
T=50, Z=60
Payoff dominant equilibrium
of CI game
20
20
50
50
Global game solution for CI
game
33.33
44.00
60.00
64.00
Risk dominant equilibrium
34.55
44.00
62.45
67.40
Maximin strategy
73.33
76.00
73.33
76.00
Unique equilibrium of PI game
32.36
41.84
60.98
66.03
Mean Threshold to Success
in sessions with CI
26.71
37.62
52.84
53.20
Mean Threshold to Success
in sessions with PI
29.73
41.83
55.33
54.04
High stake session with PI
55.97
62.69
High stake session with CI
46,91
57,15
Table 2. Theoretical equilibrium thresholds and observed mean thresholds
37
5.2. Regressions on estimated average thresholds (based on standard sessions)
We have 2 x 2 x 2 x 2 x 2 design (T, Z, location, info, order each take on 2 possible values)
Dummy variables
T
0: payoff for secure action T=20
Z
0: session with Z=100
1: T=50
1: session with
Z=60
TZ
0: if T=20 or Z=100
1: if T=50 and Z=60
1: session in
Loc(ation)
0: session in Barcelona
Frankfurt
Info(rmation)
0: session with CI
1: session with PI
1: session starting
Ord(er)
0: session starting with T=50
with T=20
1: if Order=1 and
TO
0: if Order=0 or T=20
T=50
Average threshold = b0 + b1 T + b2 Z + b3 TZ + b4 Loc + b5 Info + b6 Ord + b7 TO
Estimate parameters b0 – b7 with OLS
Average threshold = 22.6 + 27.6 T + 12.4 Z – 10.6 TZ + 1.2 Loc + 3.6 Info + 5.3 Ord – 3.5 TO
(t-values)
(10.8) (11.2)
(6.5)
(-4.2)
(1.1)
(3.8)
(3.9)
(-1.9) R2=0.91
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1. Thresholds rise in the payoff for the secure action .
2. Thresholds rise, if the hurdle to success of the risky action is increased.
- Empirical behavior reacts to changes in parameters in the same way as Laplacian belief
equilibrium.
 The concept can be used for qualitative comparative statics
3. With common information (CI) threshold tends to be smaller than with private information (PI).
- CI reduces strategic uncertainty and thereby increases the ability of a group to achieve efficient
coordination.

Public information leads to higher probability of speculative attack and
to a lower probability of inefficient liquidation of a firm.
4. In session where we started with paying 20 for the secure action both thresholds tend to be lower
than in sessions, where we started with paying 50 for this action.
- This evidence contradicts hypothesis of a numerical inertia in thresholds.
- It is consistent with numerical inertia in increments of thresholds above payoff dominance.
5. Interaction term TO reveal that order effect is stronger for T = 20 than for T = 50.
6. Location is not significant
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Statistical Analysis shows:
Threshold to crisis is 3.6 higher for CI than for PI
=>
Probability of a crisis is higher with common (public) information.
Threshold to crisis rises in T and falls in Z
As predicted by global game solution and risk dominance
=>
Interpretation:
Transaction costs and capital controls lower the probability of a crisis.
Threshold to crisis is higher (in both treatments) if we start with T=50
This contradicts hypothesis of numerical inertia in thresholds.
Explanation: Subjects decide on thresholds using a numerical increment on threshold of payoff
dominant equilibrium. Evidence supports numerical inertia in those increments.
Y* = T + d*
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7. Predictability of Crises
Theory says
Private information
Public information


unique equilibrium
multiple equilibria


attacks are predictable
attacks are not predictable
Conclusion: Public information may destabilize the economy
Experimental evidence:
In both scenarios 87% of data variation (in mean thresholds) can be explained by controlled
variables.
Average value of residuals is about the same (3.63 with CI and 3.44 with PI
Width of the interval for which attacks are not predictable is higher with private information, but
this is merely due to randomness of signals.
Predictability of average behavior is the same for both conditions
Predictability of crises is higher with public than with private information
~ predictability by an outside observer (analyst)
There is no evidence that public information might be destabilizing
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8. Testing Equilibrium theories
Hypotheses: Estimated mean individual thresholds coincide with thresholds of
a. payoff dominant equilibrium
b. risk dominant equilibrium
c. global game solution (Laplacian belief equilibrium)
d. maximin strategies
In sessions with CI all of these refinements can be rejected by 2-sided F-tests.
Observed behavior: In sessions with CI, estimated mean individual thresholds are always between
thresholds of payoff dominant equilibrium and global game solution
estimated mean
thresholds
T
Y
Global
game
solution
Maximin threshold
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Other Theories of Belief Formation
Suppose a subject believes that other subjects choose action B with some probability p.
Then the best response is to switch at some threshold Y*(p).
For p=2/3 we get
Treatment
Z=100, T=20
Z=60, T=20
Z=100, T=50
Z=60, T=50
Y*(p)
23.5
40.0
50.0
52.0
26.71
37.62
52.84
53.20
Observed mean
thresholds in CI
sessions
Predictions of p  (0.6, 0.7) cannot be rejected in sessions with CI. In sessions with PI
predictions can be rejected for any p.
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Further Results:
In sessions with common information subjects tend to coordinate within few rounds on a common
threshold that is between payoff-dominant equilibrium and Global-Game Solution.
We never observed an equilibrium that requires coordination of more than 12 out of 15 subjects.
Dispersion of estimated thresholds across sessions was the same for CI and PI
- This contradicts the theoretical prediction that with CI coordination may be subject to arbitrary selffulfilling beliefs
 There is no destabilizing effect of public information
High stake sessions yield higher thresholds:
Explanation: Risk aversion
Long sessions do not show convergence towards equilibrium, even though equilibrium is the
unique solution of iterated elimination of dominated strategies.
Explanation: Subjects deviate towards strategies that yield higher payoffs. Although not in
equilibrium, they all profit from this deviation.
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